Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs $(G,G')$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.Comment: to appear in Progress in Mathematics, Birkhause

Can (Electric-Magnetic) Duality Be Gauged?

There exists a formulation of the Maxwell theory in terms of two vector potentials, one electric and one magnetic. The action is then manifestly invariant under electric-magnetic duality transformations, which are rotations in the two-dimensional internal space of the two potentials, and local. We ask the question: can duality be gauged? The only known and battled-tested method of accomplishing the gauging is the Noether procedure. In its decanted form, it amounts to turn on the coupling by deforming the abelian gauge group of the free theory, out of whose curvatures the action is built, into a non-abelian group which becomes the gauge group of the resulting theory. In this article, we show that the method cannot be successfully implemented for electric-magnetic duality. We thus conclude that, unless a radically new idea is introduced, electric-magnetic duality cannot be gauged. The implication of this result for supergravity is briefly discussed.Comment: Some minor typos correcte

Non-Linear Realisation of the Pure N=4, D=5 Supergravity

We perform the non-linear realisation or the coset formulation of the pure N=4, D=5 supergravity. We derive the Lie superalgebra which parameterizes a coset map whose induced Cartan-Maurer form produces the bosonic field equations of the pure N=4, D=5 supergravity by canonically satisfying the Cartan-Maurer equation. We also obtain the first-order field equations of the theory as a twisted self-duality condition for the Cartan-Maurer form within the geometrical framework of the coset construction.Comment: 12 page

Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Some results on invariant theory

First published in the Bulletin of the American Mathematical Society in Vol.68 1962, published by the American Mathematical Societ

Invariant differential equations on homogeneous manifolds

First published in the Bulletin of the American Mathematical Society in Vol.83, 1977, published by the American Mathematical Societ

Radon-Fourier transforms on symmetric spaces and related group representations

First published in the Bulletin of the American Mathematical Society in Vol.71, 1965, published by the American Mathematical Societ

Wigner transform and pseudodifferential operators on symmetric spaces of non-compact type

We obtain a general expression for a Wigner transform (Wigner function) on symmetric spaces of non-compact type and study the Weyl calculus of pseudodifferential operators on them

Fundamental solutions of invariant differential operators on symmetric spaces

First published in the Bulletin of the American Mathematical Society in 1963, published by the American Mathematical Societ

Duality and Radon transform for symmetric spaces

First published in the Bulletin of the American Mathematical Society in Vol.69, 1963, published by the American Mathematical Societ